Differentiating densities on smooth manifolds
作者：
Highlights：
• Monte Carlo integration of derivatives of oscillatory functions is very expensive.
• For Lebesgue integrals, integration by parts gives rise to a density gradient g.
• g is defined as a directional derivative of density implied by a coordinate chart.
• Computation of g requires both the Jacobian and Hessian of the chart.
• First- and second-order tangent equations must be solved to find g on trajectories.
摘要
•Monte Carlo integration of derivatives of oscillatory functions is very expensive.•For Lebesgue integrals, integration by parts gives rise to a density gradient g.•g is defined as a directional derivative of density implied by a coordinate chart.•Computation of g requires both the Jacobian and Hessian of the chart.•First- and second-order tangent equations must be solved to find g on trajectories.
论文关键词：Density gradient function,Differentiable manifold,Lebesgue integral,Monte Carlo integration,Linear response
论文评审过程：Received 13 March 2021, Revised 14 May 2021, Accepted 6 June 2021, Available online 20 June 2021, Version of Record 20 June 2021.
论文官网地址：https://doi.org/10.1016/j.amc.2021.126444
